SOLUTION: Determine the domain of y= f(x) = cos^-1(3x + 2) and solve the equation y = 𝜋/4

Question 1195187: Determine the domain of y= f(x) = cos^-1(3x + 2) and solve the equation y = 𝜋/4
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part 1: Finding the domain.

Recall that the smallest cosine can get is -1, and the largest is 1
I.e. the outputs are between -1 and 1, inclusive of both endpoints

This tells us the range is

This then leads directly to the fact that the domain of

is .
The domain and range swap places when going from the original to the inverse function.
This swap is because x and y themselves swap roles.

Therefore, the input expression 3x+2 must be between -1 and 1

Subtracting 2 from all sides

Dividing all sides by 3

The inequality signs stay the same the entire time. They only would flip if we divided all sides by a negative number.

Answer: The domain as a compound inequality is
Verbally we can say x must be between -1 and -1/3, inclusive of both endpoints.
In interval notation, the domain is [-1, -1/3]. Don't forget the square brackets.

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Part 2: Solve the equation for x when y = pi/4

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